How to make an object stay inside another object?
This is a perfect case for Parenting objects.
Set your drawer as a parent to the jewels, and they'll move with it.

How to
Select the jewels, then select the drawer an set it as a parent.
Object > Parent > Object
Shortcut: Ctrl + P > Object

Objects with and without parenting.  



Further reading:
Parenting objects in Blender Manual